a.What is the optimal solution, and what is the value of the total profit contribution? b.Which constraints are binding? c.What are the dual values for the resources? Interpret each. d.If overtime can be scheduled in one of the departments, where would you recommend doing so? 6.Refer to the computer solution of the Kelson Sporting Equipment problem in Figure 3.13 (see Problem 5). a.Determine the objective coefficient ranges. b.Interpret the ranges in part (a). c.Interpret the right-hand-sides ranges. d.How much will the value of the optimal solution improve if 20 extra hours of packaging and shipping time are made available?

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Chapter 3 / Exercise 6

**An Introduction to Management Science: Quantitative Approach**

Anderson/Sweeney

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Quantitative Analysis BA 452 Supplemental Questions 310 7.Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of U.S. Oil and H shares of Huber Steel. The annual return for U.S. Oil is $3 per share and the annual return for Huber Steel is $5 per share. U.S. Oil sells for $25 per share and Huber Steel sells for $50 per share. The portfolio has $80,000 to be invested. The portfolio risk index (.50 per share of U.S. Oil and 0.25 per share for Huber Steel) has a maximum of 700. In addition, the portfolio is limited to a maximum of 1000 shares of U.S. Oil. The linear programming formulation that will maximize the total annual return of the portfolio is as follows: 𝑀𝑀𝑀𝑀𝑀𝑀3𝑈𝑈+ 5𝐻𝐻𝑀𝑀𝑀𝑀𝑀𝑀𝐶𝐶𝑀𝑀𝐶𝐶𝑀𝑀𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑀𝑀𝑡𝑡𝑀𝑀𝐶𝐶𝐶𝐶𝐶𝐶𝑀𝑀𝑡𝑡𝑟𝑟𝑠𝑠𝑡𝑡𝐶𝐶𝑟𝑟𝐶𝐶𝑠𝑠.𝑡𝑡.25𝑈𝑈+ 50𝐻𝐻 ≤80,000 𝐹𝐹𝐶𝐶𝐶𝐶𝑎𝑎𝑠𝑠𝑀𝑀𝑎𝑎𝑀𝑀𝐶𝐶𝑡𝑡𝑀𝑀𝑏𝑏𝑡𝑡𝑠𝑠0.50𝑈𝑈+ 0.25𝐷𝐷 ≤700 𝑅𝑅𝐶𝐶𝑠𝑠𝑃𝑃𝑀𝑀𝑀𝑀𝑀𝑀𝐶𝐶𝑀𝑀𝐶𝐶𝑀𝑀1𝑈𝑈≤1000 𝑈𝑈.𝑆𝑆.𝑂𝑂𝐶𝐶𝑡𝑡𝑀𝑀𝑀𝑀𝑀𝑀𝐶𝐶𝑀𝑀𝐶𝐶𝑀𝑀𝑈𝑈,𝐻𝐻 ≥0The computer solution of this problem is shown in Figure 3.14.

Quantitative Analysis BA 452 Supplemental Questions 311 a.What is the optimal solution, and what is the value of the total annual return? b.Which constraints are binding? What is your interpretation of these constraints in terms of the problem? c.What are the dual values for the constraints? Interpret each. d.Would it be beneficial to increase the maximum amount invested in U.S. Oil? Why or why not? 8.Refer to Figure 3.14, which shows the computer solution of Problem 7. a.How much would the return for U.S. Oil have to increase before it would be beneficial to increase the investment in this stock? b.How much would the return for Huber steel have to decrease before it would be beneficial to reduce the investment in this tock? c.How much would the total annual return be reduced if the U.S. Oil maximum were reduced to 900 shares?

Quantitative Analysis BA 452 Supplemental Questions 312